Jun-Hong Liang

The Wavy Ekman Layer: Langmuir Circulations, Breaking Waves, and Reynolds Stress

AbstractLarge-eddy simulations are made for the canonical Ekman layer problem of a steady wind above a uniformly rotating, constant-density ocean. The focus is on the influence of surface gravity waves: namely, the wave-averaged Stokes-Coriolis and Stokes-vortex forces and parameterized wave breaking for momentum and energy injection. The wave effects are substantial: the boundary layer is deeper, the turbulence is stronger, and eddy momentum flux is dominated by breakers and Langmuir circulations with a vertical structure inconsistent with both the conventional logarithmic layer and eddy viscosity relations. The surface particle mean drift is dominated by Stokes velocity with Langmuir circulations playing a minor role. Implications are assessed for parameterization of the mean velocity profile in the Ekman layer with wave effects by exploring several parameterization ideas. The authors find that the K-profile parameterization (KPP) eddy viscosity is skillful for the interior of the Ekman layer with wave-...

Linear and nonlinear analysis of shallow wakes

The bottom friction and the limited vertical extent of the water depth play a significant role in the dynamics of shallow wakes. These effects along with the effect of the strength of the shear layer define the wake parameter S. A nonlinear model, based on a second-order explicit finite volume solution of the depth-averaged shallow water equation in which the fluxes are obtained from the solution of the Bhatnagar–Gross– Krook (BGK) Boltzmann equation, is developed and applied to shallow wake flows for which laboratory data are available. The velocity profiles, size of the recirculating wake, oscillation frequency, and wake centreline velocity are studied. The computed and measured results are in reasonable agreement for the vortex street (VS) and unsteady bubble (UB) regimes, but not for the steady bubble (SB). The computed length of the recirculation region is about 60 % shorter than the measured value when S belongs to the SB regime. As a result, the stability investigation performed in this paper is restricted to S values away from the transition between SB and UB. Linear analysis of the VS time-averaged velocity profiles reveals a region of absolute instability in the vicinity of the cylinder associated with large velocity deficit, followed by a region of convective instability, which is in turn followed by a stable region. The frequency obtained from Koch’s criterion is in good agreement with the shedding frequency of the fully developed VS. However, this analysis does not reveal the mechanism that sets the global shedding frequency of the VS regime because the basic state is obtained from the VS regime itself. The mechanism responsible for VS shedding is sought by investigating the stability behaviour of velocity profiles in the UB regime as S is decreased towards the critical value which defines the transition from the UB to the VS. The results show that the near wake consists of a region of absolute instability sandwiched between two convectively unstable regions. The frequency of the VS appears to be predicted well by the selection criteria given in Pier & Huerre (2001) and Pier (2002), suggesting that the ‘wave-maker’ mechanism proposed in Pier & Huerre (2001) in the context of deep wakes remains valid for shallow wakes. The amplitude spectra produced by the nonlinear model are characterized by a narrow band of large-amplitude frequencies and a wide band of small-amplitude frequencies. Weakly nonlinear analysis indicates that the small amplitude frequencies are due to secondary instabilities. Both the UB and VS regimes are found to be insensitive to random forcing at the inflow boundary. The insensitivity to random noise is consistent with the linear results which show that the UB and VS flows contain regions of absolute instabilities in the near wake where the velocity deficit is large.

Langmuir turbulence in swell

AbstractThe problem is posed and solved for the oceanic surface boundary layer in the presence of wind stress, stable density stratification, equilibrium wind-waves, and remotely generated swell-waves. The addition of swell causes an amplification of the Lagrangian-mean current and rotation toward the swell-wave direction, a fattening of the Ekman velocity spiral and associated vertical Reynolds stress profile, an amplification of the inertial current response, an enhancement of turbulent variance and buoyancy entrainment rate from the pycnocline, and—for very large swell—an upscaling of the coherent Langmuir circulation patterns. Implications are discussed for the parameterization of Langmuir turbulence influences on the mean current profile and the material entrainment rate in oceanic circulation models. In particular, even though the turbulent kinetic energy monotonically increases with wave amplitude inversely expressed by the turbulent Langmuir number La, the Lagrangian shear eddy viscosity profile κ...

A Boltzmann-based finite volume algorithm for surface water flows on cells of arbitrary shapes

An explicit two–dimensional conservative finite volume model for shallow water equations is formulated and tested. The algorithm for the mass and momentum fluxes at the control surface of the finite volume is obtained from the solution of the Bhatnagar–Gross–Krook (BGK) Boltzmann equation. Unlike classical methods, BGK schemes do not require an ad–hoc splitting of advection and diffusion. The BGK scheme is second order in both time and space. The formulation of the BGK algorithm is performed for a cell of arbitrary irregular shape, but the test cases are conducted using a structured grid of quadrilateral cells. Two approximate Riemann solvers, the HLLC scheme and the two–stage Hancock–HLLC scheme, where HLL stands for Harten, Lax and van Leer and C stands for contact discontinuity, are also considered. The second–order accuracy of HLL and Hancock–HLLC schemes is obtained by MUSCL approach, where MUSCL is the acronym for Monotone Upstream–centered Schemes for Conservation Laws. The data reconstruction for all three schemes is carried out by theVan Leer limiter. The test cases involve strong shocks and expansion waves. The accuracy of the schemes are measured using an absolute error norm and a waviness error norm. The HLLC scheme is highly oscillatory for Courant number larger than 0.5, while the BGK and the Hancock–HLLC schemes are applicable for Courant numbers as high as 1.0. For a fixed value of the central processing unit (CPU) time, the absolute error of the Hancock–HLLC is slightly smaller than that of the BGK while the waviness error of the BGK is quite close to that of Hancock–HLLC. This is because (i) the Hancock–HLLC is a two–step method while the BGK is a single–step method (i.e., the Hancock–HLLC requires storage of intermediate variables, but the BGK does not), and (ii) the Hancock–HLLC schemes requires larger number of grid points than the BGK scheme for the same level of accuracy. For example, to achieve an absolute error of 0.01, the BGK requires about 600 grid points while the Hancock–HLLC requires about 800 grid points. Both the BGK and Hancock–HLLC schemes have similar convergence properties. Unlike exact or approximate Riemann solvers, BGK fluxes accounts for both waves and diffusion. The ability of the BGK scheme to model diffusion is illustrated using a viscous flow problem. Excellent agreement between the analytical and computed viscous flow solution is found. Although the BGK and Hancock–HLLC schemes perform similarly for hyperbolic problems, BGK schemes have the added advantage of being able to solve hyperbolic–parabolic problems without the need for an ad–hoc operator splitting. This is important given that the artificial splitting of advection and diffusion is known to cause artificial widening in shear layers and introduces artificial transient in regions with sharp gradients. Such problems arise when the splitting operation fails to faithfully represent the correct coupling between the physics of advection and the physics of waves.

Investigation of shallow mixing layers by BGK finite volume model

Turbulent shallow mixing layers and their associated vortical structures are ubiquitous in rivers, estuaries and coasts. Examples of these flows can be found in compound/composite channels, at the confluence of two rivers, at harbour entrances and at groyne fields. A finite volume 2D model, based on the averaging of the 3D shallow water equations with respect to depth and in which the numerical fluxes are obtained from the Bhatnagar–Gross–Krook (BGK) Boltzmann equation, is applied to shallow mixing layers for which experimental results are available. This model is hereafter referred to as the BGK model or BGK scheme. The BGK scheme is explicit, second order in time and space and conserves both mass and momentum. The BGK relaxation time is locally evaluated from the classical turbulence model of Smagorinsky. The BGK model accurately represents the mean flow field such as mean velocity profile, mean spread of the mixing layer, mean position of the mixing layer centreline and mean surface water profile. In addition, the Kelvin–Helmholtz (KH) instability including inception, vortex roll up, vortex growth by pairing and the eventual decay of the vortices by bed shear is well represented by the model. On the other hand, the magnitude of the turbulence intensity is over-predicted by the shallow water model. This discrepancy is partly due to the fact that the turbulence forcing assumed may not represent the actual random perturbations that may exist in the laboratory experiments and partly due to the inability of the depth-averaged shallow water equations to allow for the redistribution of turbulent energy along the vertical direction, since these governing equations do not model the 3D turbulence. Thus, the depth-averaged shallow water equations are well suited for investigating the KH stability and for predicting mean flow field including velocity profiles and transversal mixing of mass momentum in shallow environments. Accurate prediction of turbulence statistics would require resolving the small 3D scales with respect to water depth.

BGK based finite volume scheme for hydraulic applications

This paper explores some of the applications of Boltzmann models in the hydraulic field. A finite volume scheme with fluxes evaluated based on the solution of the Bhatnagar–Gross-Krook (BGK) Boltzmann equation for surface water flows is obtained. The consistency of the BGK equation with the classical mass and momentum equations for shallow water flow in two-lateral dimensional systems is shown. Turbulence models are incorporated through the collision time. The model is applied to a wide range of problems in surface water hydraulics including circular dam break (bore), roll waves, shallow wakes and shallow mixing layers. Where possible, model results are compared to experimental data.

A BGK model for three-dimensional shallow water problems

This paper focuses on illustrating how the Boltzmann theory can be used to formulate numerical algorithms for three-dimensional shallow water equations. The paper begins by showing that the classical three-dimensional shallow water equations are obtainable from the moments of the Boltzmann Bhatnagar-Gross-Krook (BGK) equation. This connection is then exploited to formulate a three-dimensional finite volume for shallow flows in which the fluxes are obtained on the basis of the BGK equation. The resulting scheme is illustrated using a variety of surface water problems. The advantages of using the Boltzmann-based to formulate numerical algorithms for surface water flows are summarized.